Lagrangian duality and saddle points for sparse linear programming

Abstract

The sparse linear programming (SLP) is a linear programming problem equipped with a sparsity constraint, which is nonconvex, discontinuous and generally NP-hard to solve due to the combinatorial property involved. In this paper, by rewriting the sparsity constraint into a disjunctive form, we present an explicit formula of Lagrangian dual problem for the SLP, in terms of an unconstrained piecewise-linear convex programming problem which admits a strong duality. Furthermore, we show a saddle point theorem based on the strong duality and give two first-order necessary and sufficient optimality conditions for the saddle point problem without any constraint qualification for SLP. Additionally, as an extension, we show that these main results are also valid for the case when the nonnegative box constraint $[0,u]$ in the SLP is replaced by $[-u,u]$ for any positive vector $u$.

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ORCID for Hou-Duo Qi: orcid.org/0000-0003-3481-4814

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